Let \(\Delta _n=\min _{\sigma \ne \tau \in \mathcal {I}^n}d(\sigma ,\tau )\) .
We say that the IFS \(\lbrace S_i\rbrace _{i\in \mathcal {I}}\)  satisfies the exponential separation condition (ESC) if \(\liminf _{n\rightarrow \infty }(-\log \Delta _n)/n<\infty \) .

The AWSC holds very generally, as is specified by the following result.
This is [1]}, but follows essentially from Shmerkin's result on \(L^q\) -dimensions of self-similar sets [2]}.
[[1]}]
Let \(\lbrace S_i\rbrace _{i\in \mathcal {I}}\)  be an IFS of similarities in \(\)  with contraction ratios \(\lbrace r_i\rbrace _{i\in \mathcal {I}}\) .
Let \(s\)  denote the unique solution to \(\sum _{i\in \mathcal {I}}r_i^s=1\) .
Suppose \(\lbrace S_i\rbrace _{i\in \mathcal {I}}\)  satisfies the ESC.
Then \(\lbrace S_i\rbrace _{i\in \mathcal {I}}\)  satisfies the AWSC if and only if \(s\le 1\) .

In particular, within many parametrized families of IFSs, if the similarity dimension in the parametrized family is uniformly bounded above by 1, then the AWSC holds outside of a small exceptional set (typically of Hausdorff dimension 0)—see, for example, [4]}.
In general, it is not known if the ESC assumption can be replaced with the assumption “has no exact overlaps”.
